163 lines
7.9 KiB
Matlab
163 lines
7.9 KiB
Matlab
function [LIK,lik] = gaussian_filter(ReducedForm,Y,start,DynareOptions)
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% Evaluates the likelihood of a non-linear model approximating the
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% predictive (prior) and filtered (posterior) densities for state variables
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% by gaussian distributions.
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% Gaussian approximation is done by:
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% - a Kronrod-Paterson gaussian quadrature with a limited number of nodes.
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% Multidimensional quadrature is obtained by the Smolyak operator (ref: Winschel & Kratzig, 2010).
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% - a spherical-radial cubature (ref: Arasaratnam & Haykin, 2008,2009).
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% - a scaled unscented transform cubature (ref: )
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% - Monte-Carlo draws from a multivariate gaussian distribution.
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% First and second moments of prior and posterior state densities are computed
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% from the resulting nodes/particles and allows to generate new distributions at the
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% following observation.
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% => The use of nodes is much faster than Monte-Carlo Gaussian particle and standard particles
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% filters since it treats a lesser number of particles and there is no need
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% of resampling.
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% However, estimations may reveal biaised if the model is truly non-gaussian
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% since predictive and filtered densities are unimodal.
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%
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% INPUTS
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% reduced_form_model [structure] Matlab's structure describing the reduced form model.
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% reduced_form_model.measurement.H [double] (pp x pp) variance matrix of measurement errors.
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% reduced_form_model.state.Q [double] (qq x qq) variance matrix of state errors.
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% reduced_form_model.state.dr [structure] output of resol.m.
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% Y [double] pp*smpl matrix of (detrended) data, where pp is the maximum number of observed variables.
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% start [integer] scalar, likelihood evaluation starts at 'start'.
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% smolyak_accuracy [integer] scalar.
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%
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% OUTPUTS
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% LIK [double] scalar, likelihood
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% lik [double] vector, density of observations in each period.
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%
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% REFERENCES
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%
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% NOTES
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% The vector "lik" is used to evaluate the jacobian of the likelihood.
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% Copyright (C) 2009-2013 Dynare Team
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%
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% This file is part of Dynare.
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%
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% Dynare is free software: you can redistribute it and/or modify
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% it under the terms of the GNU General Public License as published by
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% the Free Software Foundation, either version 3 of the License, or
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% (at your option) any later version.
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%
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% Dynare is distributed in the hope that it will be useful,
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% but WITHOUT ANY WARRANTY; without even the implied warranty of
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% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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% GNU General Public License for more details.
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%
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% You should have received a copy of the GNU General Public License
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% along with Dynare. If not, see <http://www.gnu.org/licenses/>.
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persistent init_flag mf0 mf1
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persistent nodes2 weights2 weights_c2 number_of_particles
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persistent sample_size number_of_state_variables number_of_observed_variables
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% Set default
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if isempty(start)
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start = 1;
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end
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% Set persistent variables.
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if isempty(init_flag)
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mf0 = ReducedForm.mf0;
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mf1 = ReducedForm.mf1;
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sample_size = size(Y,2);
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number_of_state_variables = length(mf0);
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number_of_observed_variables = length(mf1);
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number_of_particles = DynareOptions.particle.number_of_particles;
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init_flag = 1;
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end
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% compute gaussian quadrature nodes and weights on states and shocks
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if isempty(nodes2)
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if DynareOptions.particle.distribution_approximation.cubature
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[nodes2,weights2] = spherical_radial_sigma_points(number_of_state_variables);
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weights_c2 = weights2;
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elseif DynareOptions.particle.distribution_approximation.unscented
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[nodes2,weights2,weights_c2] = unscented_sigma_points(number_of_state_variables,DynareOptions);
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else
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if ~DynareOptions.particle.distribution_approximation.montecarlo
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error('Estimation: This approximation for the proposal is not implemented or unknown!')
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end
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end
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end
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if DynareOptions.particle.distribution_approximation.montecarlo
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set_dynare_seed('default');
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end
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% Get covariance matrices
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Q = ReducedForm.Q;
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H = ReducedForm.H;
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if isempty(H)
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H = 0;
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H_lower_triangular_cholesky = 0;
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else
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H_lower_triangular_cholesky = reduced_rank_cholesky(H)';
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end
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% Get initial condition for the state vector.
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StateVectorMean = ReducedForm.StateVectorMean;
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StateVectorVarianceSquareRoot = reduced_rank_cholesky(ReducedForm.StateVectorVariance)';
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state_variance_rank = size(StateVectorVarianceSquareRoot,2);
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Q_lower_triangular_cholesky = reduced_rank_cholesky(Q)';
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% Initialization of the likelihood.
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const_lik = (2*pi)^(number_of_observed_variables/2) ;
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lik = NaN(sample_size,1);
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LIK = NaN;
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SampleWeights = 1/number_of_particles ;
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ks = 0 ;
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%Estimate = zeros(number_of_state_variables,sample_size) ;
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%V_Estimate = zeros(number_of_state_variables,number_of_state_variables,sample_size) ;
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for t=1:sample_size
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% build the proposal
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[PredictedStateMean,PredictedStateVarianceSquareRoot,StateVectorMean,StateVectorVarianceSquareRoot] = ...
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gaussian_filter_bank(ReducedForm,Y(:,t),StateVectorMean,StateVectorVarianceSquareRoot,Q_lower_triangular_cholesky,H_lower_triangular_cholesky,H,DynareOptions) ;
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%Estimate(:,t) = PredictedStateMean ;
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%V_Estimate(:,:,t) = PredictedStateVarianceSquareRoot ;
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if DynareOptions.particle.distribution_approximation.cubature || DynareOptions.particle.distribution_approximation.unscented
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StateParticles = bsxfun(@plus,StateVectorMean,StateVectorVarianceSquareRoot*nodes2') ;
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IncrementalWeights = ...
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gaussian_densities(Y(:,t),StateVectorMean,...
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StateVectorVarianceSquareRoot,PredictedStateMean,...
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PredictedStateVarianceSquareRoot,StateParticles,H,const_lik,...
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weights2,weights_c2,ReducedForm,DynareOptions) ;
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SampleWeights = weights2.*IncrementalWeights ;
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SumSampleWeights = sum(SampleWeights) ;
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lik(t) = log(SumSampleWeights) ;
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SampleWeights = SampleWeights./SumSampleWeights ;
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else % Monte-Carlo draws
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StateParticles = bsxfun(@plus,StateVectorVarianceSquareRoot*randn(state_variance_rank,number_of_particles),StateVectorMean) ;
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IncrementalWeights = ...
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gaussian_densities(Y(:,t),StateVectorMean,...
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StateVectorVarianceSquareRoot,PredictedStateMean,...
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PredictedStateVarianceSquareRoot,StateParticles,H,const_lik,...
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1/number_of_particles,1/number_of_particles,ReducedForm,DynareOptions) ;
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SampleWeights = SampleWeights.*IncrementalWeights ;
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SumSampleWeights = sum(SampleWeights) ;
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%VarSampleWeights = IncrementalWeights-SumSampleWeights ;
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%VarSampleWeights = VarSampleWeights*VarSampleWeights'/(number_of_particles-1) ;
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lik(t) = log(SumSampleWeights) ; %+ .5*VarSampleWeights/(number_of_particles*(SumSampleWeights*SumSampleWeights)) ;
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SampleWeights = SampleWeights./SumSampleWeights ;
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Neff = 1/sum(bsxfun(@power,SampleWeights,2)) ;
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if (Neff<.5*sample_size && DynareOptions.particle.resampling.status.generic) || DynareOptions.particle.resampling.status.systematic
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ks = ks + 1 ;
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StateParticles = resample(StateParticles',SampleWeights,DynareOptions)' ;
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StateVectorMean = mean(StateParticles,2) ;
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StateVectorVarianceSquareRoot = reduced_rank_cholesky( (StateParticles*StateParticles')/(number_of_particles-1) - StateVectorMean*(StateVectorMean') )';
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SampleWeights = 1/number_of_particles ;
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elseif DynareOptions.particle.resampling.status.none
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StateVectorMean = (sampleWeights*StateParticles)' ;
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temp = sqrt(SampleWeights').*StateParticles ;
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StateVectorVarianceSquareRoot = reduced_rank_cholesky( temp'*temp - StateVectorMean*(StateVectorMean') )';
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end
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end
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end
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LIK = -sum(lik(start:end));
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